Constrained energy minimization and ground states for NLS with point defects

TL;DR

This paper studies the ground states of the one-dimensional nonlinear Schrödinger equation with point defects, providing explicit solutions and stability analysis for various defect models, including a novel dipole defect case.

Contribution

It introduces a new abstract concentration-compactness theorem for NLS with inhomogeneities and explicitly computes and analyzes ground states for three defect models, including the novel dipole defect.

Findings

Explicit ground states for delta, delta prime, and dipole defects.

Orbital stability of the computed ground states.

Introduction of a new theoretical framework for NLS with inhomogeneities.

Abstract

We investigate the ground states of the one-dimensional nonlinear Schr\"odinger equation with a defect located at a fixed point. The nonlinearity is focusing and consists of a subcritical power. The notion of ground state can be defined in several (often non-equivalent) ways. We define a ground state as a minimizer of the energy functional among the functions endowed with the same mass. This is the physically meaningful definition in the main fields of application of NLS. In this context we prove an abstract theorem that revisits the concentration-compactness method and which is suitable to treat NLS with inhomogeneities. Then we apply it to three models, describing three different kinds of defect: delta potential, delta prime interaction, and dipole. In the three cases we explicitly compute ground states and we show their orbital stability. This problem had been already considered for the delta and for the delta prime defect with a different constrained minimization problem, i.e. defining ground states as the minimizers of the action on the Nehari manifold. The case of dipole defect is entirely new.